Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10624 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,14,17,13,15,18,16)(19,20)(21,24)(22,25)(23,26)(27,34)(28,32)(29,31)(30,33);(4,7)(8,10)(12,15)(16,17)(19,21,22,28)(20,24,25,32)(23,27,30,29)(26,34,33,31);(3,5,6,9)(4,8,7,10)(12,16)(14,18)(15,17)(20,25)(21,27)(23,30)(24,31)(28,29)(32,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 149975);